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n-Harmonic mappings between annuli

n-Harmonic mappings between annuli,Tadeusz Iwaniec,Jani Onninen

n-Harmonic mappings between annuli   (Citations: 4)
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The central theme of this paper is the variational analysis of homeomorphisms $h\colon \mathbb X \onto \mathbb Y$ between two given domains $\mathbb X, \mathbb Y \subset \mathbb R^n$. We look for the extremal mappings in the Sobolev space $\mathscr W^{1,n}(\mathbb X,\mathbb Y)$ which minimize the energy integral \[ \mathscr E_h=\int_{\mathbb X} ||Dh(x)||^n dx. \] Because of the natural connections with quasiconformal mappings this $n$-harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal $n$-harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians, becomes truly a work of art.
Published in 2011.
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    • ...We invoke the results in [1, 4]. Within the Nitsche range (1.5) for the annuli A and A∗ the minimum is obtained (uniquely up to rotation) by the harmonic mapping...
    • ...Outside the Nitsche range (1.5) for the annuli A and A∗ the infimum of H(A, A∗) is not attained [1, 4]. It is a general fact concerning mappings...
    • ...The reader may wish to know that the nonharmonic mapping h(z) = z/|z| of A onto the unit circle is a minimizer of the Dirichlet integral [4]...

    Tadeusz Iwaniecet al. Harmonic mappings of an annulus, Nitsche conjecture and its generaliza...

    • ...integrals. Let us now look at the example in which the existence of deformations of finite conformal energy is lacking [22]...
    • ...Note that J(z,h) 0 for r 6 |z| 6 . This minimizer is actually unique up to the rotation of annuli, [22]...
    • ...! Y whose integral depends only on the homotopy class of h, regardless of its boundary values, [22, 23, 3]. The key to the proof of Theorem 1 is a sharp pointwise estimate of the stored-energy function by means of free Lagrangians, say |Dh(x)|2 + J(x,h) > L(x,h,Dh) (16)...
    • ...This is not always the case, and we have some surprise for the reader [22]...

    Tadeusz Iwaniecet al. Neohookean Deformations of Annuli

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